# What are the formulae of (1) 1 + cos2x (2) 1 - cos2x

Trigonometric identities are equations that relate different trigonometric functions and are true for all the values that are lies in their domain.

## Answer: (1) 1 + cos2x = 2cos^{2}x , (2) 1 - cos2x = 2sin^{2}x

Let us see, how to solve.

**Explanation:**

(1) Use the trigonometric formula, cos(a + b) = cos a cos b – sin a sin b and substitute a = b = x

cos(x + x) = cos x cos x – sin x sin x

cos2x = cos^{2}x - sin^{2}x

Now add 1 on both sides

1 + cos2x = 1 + cos^{2}x - sin^{2}x

Now write cos^{2}x + sin^{2}x for 1 on the right side of the equation,

1 + cos2x = cos^{2}x + sin^{2}x + cos^{2}x - sin^{2}x

= 2cos^{2}x

(2) Multiply the equation cos2x = cos^{2}x - sin^{2}x by negative 1 and add 1 on both sides.

-cos2x = - cos^{2}x + sin^{2}x

1 - cos2x = 1 - cos^{2}x + sin^{2}x

Now write cos^{2}x + sin^{2}x for 1 on the right side of the equation,

1 - cos2x = cos^{2}x + sin^{2}x - cos^{2}x + sin^{2}x

= 2sin^{2}x

### Thus, 1 + cos2x is 2cos^{2}x and 1 - cos2x is 2sin^{2}x.

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